Uncertainty Relations for Angular Momentum

TL;DR

This paper investigates the uncertainty regions for angular momentum in quantum spin systems, deriving bounds, comparing variance and entropic measures, and identifying optimal measurements across different spin values.

Contribution

It introduces a comprehensive method for characterizing uncertainty regions, derives optimal bounds, and constructs measurements minimizing worst-case uncertainties for angular momentum.

Findings

Lower bounds for uncertainty regions are established and are tight for large spins.

Optimal measurements are explicitly constructed, with output vectors approaching classical limits as spin increases.

Entropic and variance-based uncertainty bounds are compared, revealing different minimizing states for small and large spins.

Abstract

In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the "uncertainty regions'' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s goes to 1 as s tends to infinity.